The stochastic model of turbulence: Simplification of the diagram technique in high dimensions

Abstract

A stochastic model of turbulence is studied by the renormalization group approach in high dimensions d→ ∞. This asymptotic problem is of interest in connection with the fact that there are reasons to believe that in this limit the anomalous scaling vanish and Kolmogorov theory become fair. The paper shows the possibility of significant reduction in the number of Feynman diagrams in the calculation of β-function in this limit. It is shown that in addition to the previously known mechanism of simplification of calculations associated with the insignificance of diagrams that contain scalar products of different wave vectors as multipliers in the limit of d → ∞, there is a more complex mechanism, caused by the mutual reduction of certain groups of diagrams after renormalization. With these simplifications, only 9 of 4080 diagrams are significant in the thirdorder perturbation theory. This opens up additional opportunities for the calculations in the high orders of the perturbation theory. Refs 11.